nLab
Hartog's number

The Hartog's number of a cardinal numberκ\kappa is the number of ways to well-order a set of cardinality at most κ\kappa. Assuming the axiom of choice, it is the smallest ordinal number whose cardinality is greater than κ\kappa and therefore the successor of κ\kappa as a cardinal number. But even without the axiom of choice, it makes sense and is often an effective substitute for such a successor.

Definition

We will define the Hartog's number as a functorial operation from sets to well-ordered sets. The operation on numbers is just a round-about way of talking about the same thing.

So let SS be a set. Without the axiom of choice (or more precisely, the well-ordering theorem), it may not be possible to well-order SS itself, but we can certainly well-order some subsets of SS. On the other hand, if we can well-order SS (or a subset), then there may be many different ways to do so, even nonisomorphic ways. So to begin with, let us form the collection of all well-ordered subsets of SS, that is the subset of

∐A:𝒫S𝒫(A×A), \coprod_{A: \mathcal{P}S} \mathcal{P}(A \times A) ,

where ∐\coprod indicates disjoint union and 𝒫\mathcal{P} indicates power set, consisting of those pairs (A,R)(A,R) such that RR is a well-ordering. Then form a quotient set by identifying all well-ordered subsets that are isomorphic as well-ordered sets. This gives a set of well-order types, or ordinal numbers, which can itself be well-orderd by the general theory of ordinal numbers.

The Hartog's number of SS is this well-ordered set, the set of all order types of well-ordered subsets of SS. If κ\kappa is the cardinality of SS, then let κ+\kappa^+ be the cardinality or ordinal rank (as desired) of the Hartog's number of SS; this is called the Hartog's number of κ\kappa.

Even without choice, however, we can say this: If α\alpha is an ordinal number such that |α|≰κ|\alpha| \nleq \kappa, then κ+≤α\kappa^+ \leq \alpha. (Notice that we've shifted our thinking of the Hartog's number from a cardinal to an ordinal.) That is, κ+\kappa^+ is the smallest ordinal number whose cardinal number is not at most κ\kappa. This doesn't use any form of choice except for excluded middle; we only need choice to conclude that |κ+|>κ|\kappa^+| \gt \kappa.

The axiom of choice also implies the well-ordering theorem, that any set can be well-ordered. Thus with choice, κ+\kappa^+ is (now as a cardinal again) the smallest cardinal number greater than κ\kappa; this explains the notation κ+\kappa^+.

Examples

For nn a natural number regarded as the cardinal number of a finite set, n+n^+ is the usual successorn+1n + 1. This result uses excluded middle; else we get the plump successor of nn, which may be rather larger.

For ℵ0\aleph_0 the cardinality of the set of all natural numbers, the Hartog's number ℵ0+=ω1\aleph_0^+ = \omega_1 is the smallest uncountable ordinal. Assuming the axiom of choice (countable choice and excluded middle are enough), we have ℵ0+=ℵ1\aleph_0^+ = \aleph_1 as a cardinal.

In general, we get a sequence ωα\omega_\alpha of infinite cardinalities of well-orderable sets; assuming excluded middle, every infinite well-orderable cardinality shows up in this sequence. Assuming the axiom of choice, every infinite cardinal shows up, and we have |ωα|=ℵα|\omega_\alpha| = \aleph_\alpha. (Actually, there's no real need to begin with infinite cardinals; if we started with ω0=0\omega_0 = 0 instead of ω0=N\omega_0 = \mathbf{N} and ℵ0=0\aleph_0 = 0 instead of ℵ0=|N|\aleph_0 = |\mathbf{N}|, then absolutely every cardinality or well-orderable cardinality would appear.)

Revised on September 20, 2012 05:37:27
by Mike Shulman
(192.16.204.218)